Exercise 5, § 1

Assume that $k$ is infinite. Let $\mathfrak{g}$ be a Lie algebra and V a $\mathfrak{g}$-module. If $n$ is an integer $\geq 0$, denote by $V_n$ the set of $v\in V$ such that $x^nv= 0$ for all $x\in \mathfrak{g}$.

a) Show that, if $v\in V_n,x, y\in \mathfrak{g}$, then

(Use the fact that $(x+ty)^nv= 0$ for all $t\in k.$)

Replacing $y$ by $[x, y]$ in this formula, deduce$^3$ that $(x^ny-yx^n)v= 0$, and hence that $x^nyv= 0$.

b) Show that $V_n$ is a $\mathfrak{g}$-submodule of V (use $a$)$)$. In particular $V^0(\mathfrak{g}) =\bigcup_nV_n$ is a $\mathfrak{g}$-submodule of V.

c) Assume that $k=\mathbf{R}$ or $\mathbf{C}$, and denote by G a simply-connected Lie group with Lie algebra $\mathfrak{g}$; the action of $\mathfrak{g}$ on V defines a law of operation of G on V (Chap. III, §6, no. 1). Show that an element $v\in V$ belongs to $V_n$ if and only if $(s-1)^nv= 0$ for all $s\in G$; in particular $V^0(\mathfrak{g}) = V^1(G)$.